Creation Functions

Creation of Structures

GaloisRing(q, d): RngIntElt, RngIntElt -> RngGal

GR(q, d): RngIntElt, RngIntElt -> RngGal

Given integers \(q\), \(d\ge 1\), where \(q = p^a\) for prime \(p\) and \(a\ge 1\), create the default Galois ring \(\operatorname{GR}({p^a}, { d})\). The defining polynomial used to construct the ring will be that used for \({\bf F}_{p^d}\), lifted to \({\mathbb{Z}}_{p^a}\). If \(p\) is very large, it is advised to use the next function instead, because Magma must first factor \(q\) completely. The angle bracket notation can be used to assign names to the generator; e.g.: R<w> := GaloisRing(2, 3).

GaloisRing(p, a, d): RngIntElt, RngIntElt, RngIntElt -> RngGal

GR(p, a, d): RngIntElt, RngIntElt, RngIntElt -> RngGal

Check: BoolElt                    Default: true

Given a prime \(p\) and integers \(a, d\ge 1\), create the default Galois ring \(\operatorname{GR}({p^a}, { d})\). The defining polynomial used to construct the ring will be that used for \({\bf F}_{p^d}\), lifted to \({\mathbb{Z}}_{p^a}\). By default \(p\) is checked to be a strong pseudoprime for 20 random bases \(b\) with \(1 < b < p\); if the parameter Check is false, then no check is done on \(p\) at all (this is useful when \(p\) is very large and one does not wish to perform an expensive primality test on \(p\)).

GaloisRing(q, D): RngIntElt, RngUPol -> RngGal

GR(q, D): RngIntElt, RngUPol -> RngGal

Given an integer \(q\), where \(q = p^a\) for prime \(p\) and \(a\ge 1\), and a monic polynomial \(D\) over \({\mathbb{Z}}\) such that \(D\) is irreducible mod \(p\), create the Galois ring \(R = \operatorname{GR}({p^a}, { D})\). The coefficients of \(D\) are reduced modulo \(p^a\), and \(R\) is constructed to behave like the polynomial quotient ring \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\). If \(p\) is very large, it is advised to use the next function instead, because Magma must first factor \(q\) completely.

GaloisRing(p, a, D): RngIntElt, RngIntElt, RngUPol -> RngGal

GR(p, a, D): RngIntElt, RngIntElt, RngUPol -> RngGal

Check: BoolElt                    Default: true

Given a prime \(p\), an integer \(a\ge 1\), and a monic polynomial \(D\) over \({\mathbb{Z}}\) such that \(D\) is irreducible mod \(p\), create the Galois ring \(R = \operatorname{GR}({p^a}, { D})\). The coefficients of \(D\) are reduced modulo \(p^a\), and \(R\) is constructed to behave like the polynomial quotient ring \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\). The parameter Check is as above.

Names

AssignNames(~R, [f]): RngGal, [ MonStgElt ])

Procedure to change the name of the generating element in the Galois ring \(R\) to the contents of the string \(f\). When \(R\) is created, the name will be R.1. This procedure only changes the name used in printing the elements of \(R\). It does not assign to an identifier called \(f\) the value of the generator in \(R\); to do this, use an assignment statement, or use angle brackets when creating the ring. Note that since this is a procedure that modifies \(R\), it is necessary to have a reference ~R to \(R\) in the call to this function.

Name(R, 1): RngGal, RngIntElt -> RngGalElt

Given a Galois ring \(R\), return the element which has the name attached to it, that is, return the element R.1 of \(R\).

Creation of Elements

R . 1: RngGal -> RngGalElt

Generator(R): RngGal -> RngGalElt

The generator for \(R\) as an algebra over its base ring \({\mathbb{Z}}_{p^a}\). Thus, if \(R\) is viewed as \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\), then R.1 corresponds to \(x\) in this presentation.

R ! a: RngGal, RngElt -> RngGalElt

Given a Galois ring \(R\) create the element specified by \(a\); here \(a\) is allowed to be an element coercible into \(R\), which means that \(a\) may be

(i)

An element of \(R\);

(ii)

An integer, to be identified with \(a\) modulo the characteristic \(p^a\) of \(R\);

(iii)

An element of the base ring \({\mathbb{Z}}_{p^a}\) of \(R\), to be identified with the corresponding element of \(R\).

(iv)

A sequence of elements of the base ring \({\mathbb{Z}}_{p^a}\) of \(R\). In this case the element \(a_0+a_1w+\cdots+a_{n-1}w^{n-1}\) is created, where \(a=[a_0, \ldots a_{n-1}]\) and \(w\) is the generator R.1 of \(R\) over \({\mathbb{Z}}_{p^a}\).

One(R): RngGal -> RngGalElt

Identity(R): RngGal -> RngGalElt

Zero(R): RngGal -> RngGalElt

Representative(R): RngGal -> RngGalElt

These generic functions create \(1\), \(1\), \(0\), and \(0\) respectively, in any Galois ring.

Random(R): RngGal -> RngGalElt

Create a pseudo-random element of Galois ring \(R\).

Sequence Conversions

ElementToSequence(a): RngGalElt -> [ RngIntResElt ]

Eltseq(a): RngGalElt -> [ RngIntResElt ]

Given an element \(a\) of the Galois ring \(R\), return the sequence of coefficients \([a_0, \ldots, a_{n-1}]\) in the base ring \({\mathbb{Z}}_{p^a}\) of \(R\) (where \(R\) is \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\)), such that \(a=a_0+a_1w+\cdots+a_{n-1}w^{d-1}\), with \(w\) the generator of \(R\), and \(d\) the degree of \(D\).

Example: Create (ex-5d69d7)

We can define the Galois ring \(\operatorname{GR}({2^3}, { 2})\) using the default function:

> R<w> := GaloisRing(2^3, 2);
> R;
GaloisRing(2, 3, 2)

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We note that \(R\) has characteristic 8 and that \(w^2 + w + 1 = 0\) in \(R\).

> R!8;
0
> R!9;
1
> w;
w
> 4*w;
4*w
> 4*w + 4*w;
0
> w^2;
7*w + 7
> w^2 + w + 1;
0

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We can list all the elements of \(R\) by simply looping over \(R\):

> [x: x in R];
[ 0, 1, 2, 3, 4, 5, 6, 7, w, w + 1, w + 2, w + 3, w + 4, w + 5, w + 6,
    w + 7, 2*w, 2*w + 1, 2*w + 2, 2*w + 3, 2*w + 4, 2*w + 5, 2*w + 6, 2*w
    + 7, 3*w, 3*w + 1, 3*w + 2, 3*w + 3, 3*w + 4, 3*w + 5, 3*w + 6,
    3*w + 7, 4*w, 4*w + 1, 4*w + 2, 4*w + 3, 4*w + 4, 4*w + 5, 4*w +
    6, 4*w + 7, 5*w, 5*w + 1, 5*w + 2, 5*w + 3, 5*w + 4, 5*w + 5, 5*w
    + 6, 5*w + 7, 6*w, 6*w + 1, 6*w + 2, 6*w + 3, 6*w + 4, 6*w + 5,
    6*w + 6, 6*w + 7, 7*w, 7*w + 1, 7*w + 2, 7*w + 3, 7*w + 4, 7*w +
    5, 7*w + 6, 7*w + 7 ]

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We see that the elements of \(R\) can be considered as polynomials of degree at most 1, with coefficients in the range \(\{ 0 \ldots 7\}\). We can easily create elements of \(R\) also using the ! operator, and use the Eltseq function to recover the corresponding sequence of coefficients.

> R ! [1, 2];
2*w + 1
> Eltseq(2*w + 1);
[ 1, 2 ]
> Eltseq(w);
[ 0, 1 ]

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